Respuesta :
Acceleration can be broken into tangential and normal components, The tangential vector points in the direction which particle is moving and normal vector points in that direction in which curve of that object's path is turning.
Tangential component,[tex]a_{T} = \frac{r'(t) . r''(t)}{|r'(t)|}[/tex]
And normal component,[tex]a_{N} =\frac{|r'(t) X r''(t)|}{|r'(t)|}[/tex]
Given [tex]r(t)=(3+t)i+(t^{2} -2t)j[/tex]
= < 3+t , [tex]t^{2} -2t[/tex] >
r'(t) = <1 , 2t-2>
r''(t) = <0,2>
Let's plugin these values in [tex]a_{T}[/tex] and [tex]a_{N}[/tex]
[tex]a_{T} = \frac{<1,2t-2> . <0,2>}{|<1,2t-2>|}[/tex]
= [tex]\frac{(2t-2)2}{\sqrt{1+(2t-2)^{2} } }[/tex]
=[tex]\frac{4(t-1)}{\sqrt{4t^{2}-8t+5 } }[/tex]
[tex]a_{N}=\frac{|<1,2t-2>X<0,2>|}{|<1,2t-2|>} = \frac{|<0,2>|}{\sqrt{4t^{2}-8t+5} }[/tex]
= [tex]\frac{2}{\sqrt{4t^{2}-8t+5 } }[/tex]
The tangential acceleration component and the normal acceleration component are [tex]a_t = \dfrac{4(t-1)}{\sqrt{4t^2 - 4t + 5}}[/tex] and [tex]a_n = \dfrac{2}{\sqrt{4t^2 - 4t + 5}}[/tex]
What is acceleration?
The acceleration is the rate of change of the velocity is known as acceleration. It is a vector quantity. And its SI unit is a meter per second square.
Given
[tex]\rm r(t) = (3+t)i + (t^2 - 2t)j[/tex] is an acceleration expression.
Then
[tex]\rm r'(t) = 3i + (2t - 2)j[/tex]
And
[tex] \rm r"(t) = 2j [/tex]
The tangential acceleration component will be.
[tex] \rm a_t = \dfrac{r'(t).r"(t)}{\left|r'(t) \right|} \\ a_t = \dfrac{[3i + (2t -2)j].[2j]}{\left| 3i + (2t-2)j \right|} \\ a_t= \dfrac{4(t-1)}{\sqrt{1^2 + (2t - 2)^2}}\\ a_t = \dfrac{4(t-1)}{\sqrt{4t^2 - 4t + 5}}[/tex]
The normal acceleration component will be.
[tex] \rm a_n = \dfrac{r'(t) x r"(t)}{\left|r'(t) \right|} \\ a_n = \dfrac{[3i + (2t -2)j] x [2j]}{\left| 3i + (2t-2)j \right|} \\ a_n= \dfrac{2}{\sqrt{1^2 + (2t - 2)^2}}\\ a_n = \dfrac{2}{\sqrt{4t^2 - 4t + 5}}[/tex]
Thus, the tangential acceleration component and the normal acceleration component are [tex]a_t = \dfrac{4(t-1)}{\sqrt{4t^2 - 4t + 5}}[/tex] and [tex]a_n = \dfrac{2}{\sqrt{4t^2 - 4t + 5}}[/tex]
More about the acceleration link is given below.
https://brainly.com/question/2437624
